# Find the x and y intercepts of a rational function

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So ladies and gentlemen of course when we want to find the horizontal and vertical vertical asymptotes you guys need to remember where did these horizontal and vertical asymptotes what they tell us what do we know at each point all right when we have remember now were talking about functions. So this is our f of x. And thats our x.
That looks like a y. So when were dealing with this and i have an i say what is my and im just losing. It arent i okay so when i have my x intercept.
That means when the graph crosses the x axis right what is my value of my f of x. My output at this lets say heres the intercept. What is the value of my of x at that point zero f of x.
Equals zero at this point right. Because youre not going up or down at all so. When i say x intercept.
Now. I want to find the x. Intercept of a function.
Either my app of x. Equals zero or sometimes we might say y equals. Zero for just dealing with linear lines.
And not functions. But x. Intercept your output at x.
Intercept your output is equal to zero. So what are you putting for zero. You put zero in for f of x 0.
Equals. 5 plus 2x divided by 1 plus. X.
That was for finding the vertical asymptotes which we did last video now what were doing is were trying were trying to find the x and y intercepts okay. So this is good so for the vertical for the asymptotes right thats different than an intercept. The asymptotes is where your graph is going to approach vertically.
But now were just finding the intercepts where it crosses the x intercepts okay.

So now i got a solve for 0. Well. How do you how do you solve a rational function like this well the first thing is always to get whatevers on your bottom off your bottom right.
So how do we get 1 plus x. Off the bottom. We can multiply.
It by both on both sides. So therefore im left with 0 equals. 5 plus 2x all right subtract.
5. Relax divide by two so therefore x equals negative five house. Which it is what about negative two point five.
Okay so we could say negative five halves comma zero. Thats your x intercept. Okay.
Thats where the graph. I dont know what the graph looks like you can do new york graphing calculator. But the graph i know crosses at negative five pass because i did my work there okay now so if f of x.
Equals. Zero is for your x intercept. What do you think equals zero for your y intercept.
The x. Right. So you just write out y intercept.
Is when x equals zero. So what do we do just put in 0. For x.
So you do f of x. Equals. 5 plus 2 times 0 divided by 1 plus 0.
So f of x equals.

5. So that means my y intercept or my f of x. Intercept.
Is going to equal 1 2. 3. 4 or 5.
So my graph crosses there and it crosses. There now its not a line you know connected thats what we do in algebra. 2.
Thats what im working with them theyre finding the x y intercepts and connecting it right. But we have some graph that we need to look at and see what its going to look like so we go back to our intercepts which we did last video. Im not gonna make you guys graph it all right.
But i just want to show you guys how to find your solution points so x equals negative. 1. Thats your horizontal asymptote correct or is on stokes is negative.
1. So that means your graph approaches this line. Okay it approaches that line.
So i dont know how its going to open like this approaches that way if remember it has to approach infinity or negative infinity for it to be an asymptote so the next thing. It says is choose some solution points. And i need you guys to show me that their solution points.
So what i like you guys to do is on each side of the at some point. When i ask you to find your solution points for each side of at some point. Id like you to choose two points.
Now which two points. Do you choose this always gets students confused. Its arbitrary pick whatever points you want to 0.
What zero you already picked which was your y intercept. Id like you to show me two other points all right so lets make it easy on ourselves. So what ill do is theyll do an xy table.
And ill pick why dont we pick one and two right and then to the left of this.

Why dont we do negative. One and negative three all right so pick two points to the left and two points to the right that do not include your y intercept sorry yeah. We cant do that im sorry im at negative two youre gonna want to pick yes and your homework that weve talked about solution points.
So i want you guys to go back. And do the solution points okay. So you look at your vertical asymptote and you just pick the points to the left and to the right of your asymptote.
Okay. So i just want you to show me that theres two of it yeah exactly so the graph code was gonna help you know what the graphs can look like but i still want you good. So lets just do these this is pretty easy mathematically for us to go through.
So. Lets put in 1 2. Times.
1. Is walt 2 2. Plus or 2 plus.
5 is 7. So its gonna be 7. Halves right okay.
Im gonna give to thank you so for the next. One. Lets do 2 2 times.
2. Is 4 4 plus. 5 is 9.
So thats going to be 9 thirds lets do negative 2 2. Times negative. 2.
Is negative 4 5. Plus negative. 4.
Is 1 and thats going to be 1 over negative 1.

Which is negative 1 lets do negative 3 2 times negative 3 is negative 6 negative 6 plus 5 is negative 1 so its negative 1 over negative 2. Which is going to be negative 1 over negative 2 which would be a positive. 3 halves.
So lets take a look at this if i go over to 1. Thats going to be 2 or 3. And a half right as you write this kind of decimal yes 9.
3. Is actually obviously. 3.
Right so. 7. Halves is goodness so i go over 1.
Im going to go to three point five one two three point five and then i go to nine thirds which is or at one two positive two. Im going up to three all right then we have negative two which goes to negative one and then at negative three one two three. Im up at one half.
Okay now im not asking you guys to graph the line. However if you use your graphing. Utility and you use your y intercept or your horizontal asymptote.
Im sorry. Which looks like this you can actually shape the graph to look something like this is it okay. But im not gonna ask you guys to graph it all right all i want all im concerned about right.
Now is that you can find the x and y intercepts and show. Me you can do that and that you can pick some solution points to plug in so whats your points. You pick again just pick points to left and to right of your asymptote.
Yes. Just make. The table and figure out the solution points.
Yes. Yes yes. Its it depends on action points.
Is what when your plug damon no you can just show me that your points. I just took my vertical a spill and i picked two points to left in two points to the right i just pick those points. I knew you made you could pick what everyone yeah two points the right yep yes.
No you do not have to .